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Mirrors > Home > MPE Home > Th. List > trinxp | Structured version Visualization version GIF version |
Description: The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a Cartesian square is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.) |
Ref | Expression |
---|---|
trinxp | ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpidtr 5982 | . 2 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) | |
2 | trin2 5983 | . 2 ⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)) → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴))) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∩ cin 3935 ⊆ wss 3936 × cxp 5553 ∘ ccom 5559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-co 5564 |
This theorem is referenced by: psss 17824 |
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